What's the best way to rotate objects identically in their own direction?
It seems to work fine:

Select the stick called Cylinder.002. Rotate it as much as you want on the X axis, move it until it's correctly placed:



Delete the other sticks, select 002, press AltD to link-duplicate, in the Transform Pivot Point menu choose 3D Cursor, rotate 120° on the Z axis:



Do it again.

If you need to animate the sticks, first parent the initial stick to an empty, then duplicate both the empty and the stick, move and rotate, and you'll animate each stick: